\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip:

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example 0.0.1

Let \(Q\) be a manifold. The cotangent bundle \(X=T^*Q\) is an exact symplectic manifold; call the primitive \(\lambda_{can}=\sum_{i=1}^np_idq_i\). In local coordinates, the Liouville vector field is \(Z= \sum_{i=1}^n p_i \partial p_i\). The flow of this vector field acts by scalar multiplication on the fibers of \(T^*Q\), \begin{align*}\phi^t: T^*Q\to T^*Q && (q, p) \mapsto (q, tp)\end{align*} which inflates the symplectic form. Now equip \(Q\) with a metric \(g\). The metric determines a unit sphere bundle \(S^*Q\subset T^*Q\), which is transverse to \(V\). Letting \(\alpha=\lambda|_{S^*Q}\) makes \((S^*Q, \alpha)\) a contact manifold. The time \(t\)-flow on \((S^*Q, \alpha)\) corresponds to the time \(t\) geodesic flow on the base ((geodesics and symplectic cohohomology of the cotangent bundle)). It is known ([Fet52]) that every closed manifold has at least 1 closed geodesic, so all of these examples of contact manifolds have a Reeb orbit.

References

[Fet52]Abram Il'ich Fet. Variational problems on closed manifolds. Matematicheskii Sbornik, 72(2):271--316, 1952.